Season chicken generously with salt and pepper. Chill, covered, overnight.
Let chicken stand at room temperature for 1 hour. Preheat oven to 450°F. Toss carrots and shallots with 2 tablespoons oil and season with salt and pepper; arrange in an even layer in the center of a large roasting pan to create a rack for the chicken. Pat chicken dry; rub with remaining 1 tablespoon oil, tie legs with twine, and place over vegetables in roasting pan. Add garlic and lemon halves to pan. Roast 25 minutes.
Reduce oven temperature to 375°F; cook, flipping vegetables once, until vegetables are tender and chicken is golden brown and a meat thermometer inserted into the the thickest part of the breast registers 155°F, 30 to 40 minutes. Transfer carrots and shallots to a serving platter; squeeze with lemon juice. Transfer chicken to a cutting board to rest.
Squeeze garlic cloves from skins and mash into a paste. Toss with bread cubes in pan juices and season lightly with salt and pepper; spread into an even layer. Bake, tossing once, until golden and crisp, 9 to 12 minutes.
To serve, carve chicken and arrange on platter. Scatter croutons around chicken and serve sprinkled with parsley.